Exponential distributions are frequently used for modeling the amount of time that passes until a specific event occurs. For example, exponential distributions could be used to model the time between two earthquakes, the amount of delay between internet packets, or the amount of time a piece of machinery can run before needing repair.
Details
We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.
In the following, let \(X\) be an Exponential random variable with
rate parameter rate = \(\lambda\).
Support: \(x \in (0, \infty)\)
Mean: \(\frac{1}{\lambda}\)
Variance: \(\frac{1}{\lambda^2}\)
Probability density function (p.d.f):
$$ f(x) = \lambda e^{-\lambda x} $$
Cumulative distribution function (c.d.f):
$$ F(x) = 1 - e^{-\lambda x} $$
Moment generating function (m.g.f):
$$ \frac{\lambda}{\lambda - t}, for t < \lambda $$
Examples
set.seed(27)
X <- Exponential(5)
X
#> [1] "Exponential(rate = 5)"
mean(X)
#> [1] 0.2
variance(X)
#> [1] 25
skewness(X)
#> [1] 2
kurtosis(X)
#> [1] 6
random(X, 10)
#> [1] 0.01161126 0.28730930 1.15993941 0.29660927 0.38431337 0.04643808
#> [7] 0.06969554 0.10900366 0.50608948 0.03759968
pdf(X, 2)
#> [1] 0.0002269996
log_pdf(X, 2)
#> [1] -8.390562
cdf(X, 4)
#> [1] 1
quantile(X, 0.7)
#> [1] 0.2407946
cdf(X, quantile(X, 0.7))
#> [1] 0.7
quantile(X, cdf(X, 7))
#> [1] 6.989008